ecap: R Tree Diagram he first branch separates the primary event into two (or more) possible outcomes T The second sets of branches are the outcomes for each of the conditions. They are conditional probabilities. Joint probabilities are at the end of the branches. Random Variable (RV) is a Quantitative (Q) variable whose value depends upon some A random event or on a random outcome of an experiment (i.e. chance). Use Capital Letter: X Denote the Probability of a certain outcome as P(X = x) andom Experiments: R A probability distribution is a table (or a formula) giving the possible values (or intervals) of a RV and. The graph of a probability distribution is called a probability histogram robability Distribution Characteristics: P A distribution of a probability is true if the x's are mutually exclusive and exhaustive, each value of X is between 0 and 1, and sum of probabilities is 1 Mean of a Random Variable =Σxi p(x)) The mean is a weighted average of the xi, The values are weighted by their probabilities Standard Deviation of a Random Variable ean of a RV = Expected Value of X M The mean of a Random Variable X is also called the Expected Value of X , denoted E(X) The EV (or mean) of x describes its long-run average outcome. The larger the number of trials, the closer the average will be to μ trial is a Bernoulli trial if there are only 2 possible outcomes from each trial. A Called “Success” and “Failure”, trials are independent. The probability of getting a “Success” remains constant from trial to trial A common example is flip a coin. X will be a Binomial Variable if it results from repeated Bernoulli trials of an experiment ormula F n=number of bernoulli trials The Random Variable X = # of ‘successes ote: the n in (1-P)^n-x should be 1: so (1-P)^1-x N n is how many total combinations or total number of trails there are and x is number of trails that are successes t Boston Logan Airport, 84% of the flights in Sep 2023 arrived on time (or early). A 1. What is the random variable of interest? What number of flights arrived on time 2. Why is it a binomial variable? Because for each individual flight there are two options: arrive on time or don’t arrive on time, and assuming that arriving or not doesn’t affect other flights. 3. What is the probability of ‘success’? As the question says 84% of the time flights are arriving. 4. What is the probability flights arrival on time 12 times in a row P(X=12) = .84^12 =0.123410307 5. What is the probability that exactly 10 were on time? So what about the second half the *(1-p)^n-x e didn’t use it because the question didn’t ask for a delay. So like 3 arrivals then one delay W would use it. But now we can: P(X=10)=(0.84^10)*(1-0.84)^(1-0.84)= 0.130452395 6. Would it be unusual for fewer than three of the flights to be on time? ind prob of 0 success, 1 success, 2, success, 3 success, add them and if its very small then it F could be unusual Mean and SD of a Binomial Random Variable If 84% of all flights are on time, 7. Find the Expected number (mean number) of on-time flights in a sample of 12 flights 12*.84=10.08 8. Find the standard deviation Use formula above requency Distribution and Probability distribution and Sampling Distribution: • Show how F observed or theoretical data behave Histograms(bars) and Smooth curve can be discrete or continuous ontinuous Distributions C Often described by a formula rather than a table Probability graph is a continuous curve which we call the Probability Density Function (PDF) roperties: The total area under the curve is equal to 100% or 1 P The probability of something occurring is the portion of the area under the entire curve That ism the probability is equal to the proportion of the population that falls within any given interval s you can see while it is a bit skewed: you can tell that it still follows the bell curve somewhat A or a Normal Distribution. Under certain conditions Conditions: Symmetric • Bell-shaped • Mean = Median = Mode • Extends indefinitely in both directions, approaching, but never touching, the horizontal axis • Completely described by its mean and standard deviation ny Normal Distribution can be A transformed into the Standard Normal Distribution by standardizing the observations into a z-score!
